Abstract
The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.
Highlights
The existence and uniqueness of solutions to stochastic differential equations driven by Brownian motion have been studied by many authors
The non-local coupled system was studied by some authors
The results are important since they cover non-local generalizations of fractional stochastic differential equations (FSDE), more applications are arising in fields such as heat conduction, electromagnetic theory and dynamic system
Summary
The existence and uniqueness of solutions to stochastic differential equations driven by Brownian motion have been studied by many authors (see [1,2,3] ). The non-local coupled system was studied by some authors On the stochastic fractional operators and the solution of non-local coupled systems of stochastic differential equations see [7,16]. We study the existence of solutions of a coupled system of Itô-differential equation and arbitrary (fractional)orders random differential equation subject to two coupled systems of non-local random and stochastic integral conditions. On Xo and Yo , h1 and h2 and the solution Y ∈ C ( I, L2 (Ω)) on D α X (t) will be studied
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